John Keats wrote this poem in reaction to having read "The Iliad" and/or "The Odyssey" by Homer. In the first 4 lines, the poet says that he has often been to the "realms of gold/...Which bards in fealty to Apollo hold." What he means is that he has read the Greek myths and even has read a different translation of Homer's writings:

Oft of one wide expanse had I been told

That deep-brow'd Homer ruled as his demesne;

But it wasn't until he read Chapman's translation that the story really came alive to him.

Now, to interpret the line you are asking about. By "deep-brow'd Homer," Keats is referring to a sculpted bust of the poet. I've pasted a link to a site where you can see what that bust looks like. Homer is "deep-brow'd," or wrinkled, presumably from deep thoughts. The word "demesne" is an old, archaic term meaning "domain." So Keats is saying that Homer ruled these stories as his domain; they were his territory.

John Keats(1795-1821) the English Romantic poet was the son of an ostler-a person who looked after the horses in a stable. He had a rudimentary school education and never went to university. Nevertheless, he was fascinated by ancient Greek classical poetry. The only way he could read Homer's epics was in an English translation.

George Chapaman (1559-1634) the Elizabethan poet and dramatist had translated Homer's epics into English. In October 1816 Keats read this translation throughout the night and then wrote the sonnet "On First Looking Into Chapman's Homer." The sonnet expresses Keats' intense joy and amazement on reading the great epics of Homer in English for the first time.

"deep-browed" refers to the intellect of Homer. Homer's "Iliad" and "Odyssey" are the seminal works of western literature which continue to influence writers even today.

"demesne" is a medieval word meaning 'domain.' The word reveals Keats' love for all things medieval and colours the poem with an archaic tinge.

Keats,the reader and lover of poetry, compares himself to an explorer who has travelled far and wide and that in the course of his voyages he has heard of Homer's famous 'domain' but that he couldn't visit it and experience its beauties till he had read the English translation of Chapman.